Kobayashi pseudometric on hyperkähler manifolds

Journal of London Mathematical Society. 2014. Vol. 90. No. 2. P. 436–450.

Kamenova L., Lu S., Verbitsky M.

The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi–Yau manifolds. Using ergodicity of complex structures, we prove this for all hyperkähler manifold with b_2\geqslant 7 that admits a deformation with a Lagrangian fibration and whose Picard rank is not maximal. The Strominger-Yau-Zaslow (SYZ) conjecture claims that parabolic nef line bundles on hyperkähler manifolds are semi-ample. We prove that the Kobayashi pseudometric vanishes for any hyperkähler manifold with b_2\geqslant 7 if the SYZ conjecture holds for all its deformations. This proves the Kobayashi conjecture for all K3 surfaces and their Hilbert schemes.

Keywords: complex manifold, hyperkahler manifolds, Гиперкэлерово многообразие, Lagrangian fibration, Kobayashi metric, комплексное многообразие, метрика Кобаяши, лагранжево слоение